Μεταβατικότητα
Μεταβατικότης Transitivity, Μεταβατική Ιδιότητα [[image:Numbers-03-goog.jpg|thumb|300px| Διακριτά Μαθηματικά Αριθμητική Αριθμοθεωρία Αριθμός Τελεστής ---- Αλγεβρικές Πράξεις Πρόσθεση Αφαίρεση Πολλαπλασιασμός Διαίρεση ---- Συνολοϊκές Πράξεις Συνολοϊκή Ένωση Συνολοϊκή Τομή ---- Λογικές Πράξεις Σύζευξη (Conjunction) Διάζευξη (Disjunction) Άρνηση (Negation) ---- Ιδιότητες Πράξεων Ανακλαστική Ιδιότητα Αντιμεταθετική Ιδιότητα Προσεταιριστική Ιδιότητα Επιμεριστική Ιδιότητα ]] - Μία Αλγεβρική Ιδιότητα. Ετυμολογία Η ονομασία "Μεταβατικότητα" σχετίζεται ετυμολογικά με την λέξη "μετάβαση". Εισαγωγή In mathematics, a binary relation R'' over a set ''X is transitive if * whenever an element a'' is related to an element ''b, and * b'' is in turn related to an element ''c, then a'' is also related to ''c. Transitivity (or transitiveness) is a key property of both partial order relations and equivalence relations. Formal definition In terms of set theory, the transitive relation can be defined as: : \forall a,b,c \in X: (aRb \wedge bRc) \Rightarrow aRc Examples For example, "is greater than", "is at least as great as," and "is equal to" (equality) are transitive relations: : whenever A > B and B > C, then also A > C : whenever A ≥ B and B ≥ C, then also A ≥ C : whenever A = B and B = C, then also A = C. On the other hand, "is the mother of" is not a transitive relation, because if *Alice is the mother of Brenda, and *Brenda is the mother of Claire, then *Alice is not the mother of Claire. What is more, it is antitransitive: *Alice can never be the mother of Claire. Then again, in biology we often need to consider motherhood over an arbitrary number of generations: the relation "is a matrilinear ancestor of". This is a transitive relation. More precisely, it is the transitive closure of the relation "is the mother of". More examples of transitive relations: * "is a subset of" (set inclusion) * "divides" (divisibility) * "implies" (implication) Properties Closure properties The converse of a transitive relation is always transitive: *e.g. knowing that "is a subset of" is transitive and "is a superset of" is its converse, we can conclude that the latter is transitive as well. The intersection of two transitive relations is always transitive: *knowing that "was born before" and "has the same first name as" are transitive, we can conclude that *"was born before and also has the same first name as" is also transitive. The union of two transitive relations is not always transitive. For instance *"was born before or has the same first name as" is not generally a transitive relation. The complement of a transitive relation is not always transitive. For instance, *while "equal to" is transitive, "not equal to" is only transitive on sets with at most one element. Other properties A transitive relation is asymmetric if and only if it is irreflexive. Lemma 1.1 (iv). Note that this source refers to asymmetric relations as "strictly antisymmetric". Properties that require transitivity * Preorder – a reflexive transitive relation * Partial order – an antisymmetric preorder * Total preorder – a total preorder * Equivalence relation – a symmetric preorder * Strict weak ordering – a strict partial order in which incomparability is an equivalence relation * Total ordering – a total, antisymmetric transitive relation Counting transitive relations No general formula that counts the number of transitive relations on a finite set is known.Steven R. Finch, "Transitive relations, topologies and partial orders", 2003. However, there is a formula for finding the number of relations that are simultaneously reflexive, symmetric, and transitive – in other words, equivalence relations – , those that are symmetric and transitive, those that are symmetric, transitive, and antisymmetric, and those that are total, transitive, and antisymmetric. PfeifferGötz Pfeiffer, "Counting Transitive Relations", Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2. has made some progress in this direction, expressing relations with combinations of these properties in terms of each other, but still calculating any one is difficult. See also.Gunnar Brinkmann and Brendan D. McKay,"Counting unlabelled topologies and transitive relations" Υποσημειώσεις Εσωτερική Αρθρογραφία * Ανακλαστική Ιδιότητα * Μεταθετική Ιδιότητα * Μεταβατική Ιδιότητα *Προσεταιριστική Ιδιότητα *Επιμεριστική Ιδιότητα *Αντιμεταθετική Ιδιότητα * Transitive closure * Transitive reduction * Intransitivity * Reflexive relation * Symmetric relation * Antisymmetric relation * Quasitransitive relation * Nontransitive dice Βιβλιογραφία *Ralph P. Grimaldi, Discrete and Combinatorial Mathematics, . *Gunther Schmidt, 2010. Relational Mathematics. Cambridge University Press, . Ιστογραφία *Ομώνυμο άρθρο στην Βικιπαίδεια *Ομώνυμο άρθρο στην Livepedia * Transitivity in Action at cut-the-knot *[ ] Category: Αλγεβρικές Ιδιότητες